Boltzmann Transport Equation for Particle Transport

1. Boltzmann Transport Equation for Particle Transport Distribution Function of Particles: f= f(r,p,t) --probability of particle occupation of momentum p at location rand time t Equilibrium Distribution: f0, i.e. Fermi-Dirac for electrons, Bose-Einstein for phonons Non-equilibrium, e.g. in a high electric field or temperature gradient: Relaxation Time Approximation t Relaxation time

2. Energy Flux q v Energy flux in terms of particle flux carrying energy: dk q k f Vector Integrate over all the solid angle: Scalar Integrate over energy instead of momentum: Density of States: # of phonon modes per frequency range

3. Continuum Case BTE Solution: Quasi-equilibrium Direction x is chosen to in the direction of q Energy Flux: Fourier Law of Heat Conduction: t(e) can be treated using Callaway method (Phys. Rev. 113, 1046) If v and t are independent of particle energy, e, then  Kinetic theory:

4. At Small Length/Time Scale (L~l or t~t) Define phonon intensity: From BTE: 0 Equation of Phonon Radiative Transfer (EPRT) (Majumdar, JHT 115, 7): Heat flux: Acoustically Thin Limit (L<<l) and for T << qD Acoustically Thick Limit (L>>l)

5. Outline • Macroscopic Thermal Transport Theory – Diffusion • -- Fourier’s Law • -- Diffusion Equation • Microscale Thermal Transport Theory – Particle Transport • -- Kinetic Theory of Gases • -- Electrons in Metals • -- Phonons in Insulators • -- Boltzmann Transport Theory •  Thermal Properties of Nanostructures • -- Thin Films and Superlattices • -- Nanowires and Nanotubes • -- Nano Electromechanical System

6. Thin Film Thermal Conductivity Measurement 3w method (Cahill, Rev. Sci. Instrum. 61, 802) Metal line Thin Film L 2b V • I~ 1w • T ~ I2 ~ 2w • R ~ T ~ 2w • V~ IR ~3w I0 sin(wt) Substrate

7. Silicon on Insulator (SOI) Ju and Goodson, APL 74, 3005 IBM SOI Chip Lines: BTE results Hot spots!